Optimal. Leaf size=205 \[ \frac{910}{3} a^{12} b^3 x^{3/2}+\frac{1365}{2} a^{11} b^4 x^2+\frac{6006}{5} a^{10} b^5 x^{5/2}+\frac{5005}{3} a^9 b^6 x^3+\frac{12870}{7} a^8 b^7 x^{7/2}+\frac{6435}{4} a^7 b^8 x^4+\frac{10010}{9} a^6 b^9 x^{9/2}+\frac{3003}{5} a^5 b^{10} x^5+\frac{2730}{11} a^4 b^{11} x^{11/2}+\frac{455}{6} a^3 b^{12} x^6+\frac{210}{13} a^2 b^{13} x^{13/2}+105 a^{13} b^2 x+30 a^{14} b \sqrt{x}+a^{15} \log (x)+\frac{15}{7} a b^{14} x^7+\frac{2}{15} b^{15} x^{15/2} \]
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Rubi [A] time = 0.111991, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{910}{3} a^{12} b^3 x^{3/2}+\frac{1365}{2} a^{11} b^4 x^2+\frac{6006}{5} a^{10} b^5 x^{5/2}+\frac{5005}{3} a^9 b^6 x^3+\frac{12870}{7} a^8 b^7 x^{7/2}+\frac{6435}{4} a^7 b^8 x^4+\frac{10010}{9} a^6 b^9 x^{9/2}+\frac{3003}{5} a^5 b^{10} x^5+\frac{2730}{11} a^4 b^{11} x^{11/2}+\frac{455}{6} a^3 b^{12} x^6+\frac{210}{13} a^2 b^{13} x^{13/2}+105 a^{13} b^2 x+30 a^{14} b \sqrt{x}+a^{15} \log (x)+\frac{15}{7} a b^{14} x^7+\frac{2}{15} b^{15} x^{15/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{15}}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (15 a^{14} b+\frac{a^{15}}{x}+105 a^{13} b^2 x+455 a^{12} b^3 x^2+1365 a^{11} b^4 x^3+3003 a^{10} b^5 x^4+5005 a^9 b^6 x^5+6435 a^8 b^7 x^6+6435 a^7 b^8 x^7+5005 a^6 b^9 x^8+3003 a^5 b^{10} x^9+1365 a^4 b^{11} x^{10}+455 a^3 b^{12} x^{11}+105 a^2 b^{13} x^{12}+15 a b^{14} x^{13}+b^{15} x^{14}\right ) \, dx,x,\sqrt{x}\right )\\ &=30 a^{14} b \sqrt{x}+105 a^{13} b^2 x+\frac{910}{3} a^{12} b^3 x^{3/2}+\frac{1365}{2} a^{11} b^4 x^2+\frac{6006}{5} a^{10} b^5 x^{5/2}+\frac{5005}{3} a^9 b^6 x^3+\frac{12870}{7} a^8 b^7 x^{7/2}+\frac{6435}{4} a^7 b^8 x^4+\frac{10010}{9} a^6 b^9 x^{9/2}+\frac{3003}{5} a^5 b^{10} x^5+\frac{2730}{11} a^4 b^{11} x^{11/2}+\frac{455}{6} a^3 b^{12} x^6+\frac{210}{13} a^2 b^{13} x^{13/2}+\frac{15}{7} a b^{14} x^7+\frac{2}{15} b^{15} x^{15/2}+a^{15} \log (x)\\ \end{align*}
Mathematica [A] time = 0.0689886, size = 205, normalized size = 1. \[ \frac{910}{3} a^{12} b^3 x^{3/2}+\frac{1365}{2} a^{11} b^4 x^2+\frac{6006}{5} a^{10} b^5 x^{5/2}+\frac{5005}{3} a^9 b^6 x^3+\frac{12870}{7} a^8 b^7 x^{7/2}+\frac{6435}{4} a^7 b^8 x^4+\frac{10010}{9} a^6 b^9 x^{9/2}+\frac{3003}{5} a^5 b^{10} x^5+\frac{2730}{11} a^4 b^{11} x^{11/2}+\frac{455}{6} a^3 b^{12} x^6+\frac{210}{13} a^2 b^{13} x^{13/2}+105 a^{13} b^2 x+30 a^{14} b \sqrt{x}+a^{15} \log (x)+\frac{15}{7} a b^{14} x^7+\frac{2}{15} b^{15} x^{15/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 164, normalized size = 0.8 \begin{align*} 105\,{a}^{13}{b}^{2}x+{\frac{910\,{a}^{12}{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{\frac{1365\,{a}^{11}{b}^{4}{x}^{2}}{2}}+{\frac{6006\,{a}^{10}{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{3}}{3}}+{\frac{12870\,{a}^{8}{b}^{7}}{7}{x}^{{\frac{7}{2}}}}+{\frac{6435\,{a}^{7}{b}^{8}{x}^{4}}{4}}+{\frac{10010\,{a}^{6}{b}^{9}}{9}{x}^{{\frac{9}{2}}}}+{\frac{3003\,{a}^{5}{b}^{10}{x}^{5}}{5}}+{\frac{2730\,{a}^{4}{b}^{11}}{11}{x}^{{\frac{11}{2}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{6}}{6}}+{\frac{210\,{a}^{2}{b}^{13}}{13}{x}^{{\frac{13}{2}}}}+{\frac{15\,a{b}^{14}{x}^{7}}{7}}+{\frac{2\,{b}^{15}}{15}{x}^{{\frac{15}{2}}}}+{a}^{15}\ln \left ( x \right ) +30\,{a}^{14}b\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956984, size = 220, normalized size = 1.07 \begin{align*} \frac{2}{15} \, b^{15} x^{\frac{15}{2}} + \frac{15}{7} \, a b^{14} x^{7} + \frac{210}{13} \, a^{2} b^{13} x^{\frac{13}{2}} + \frac{455}{6} \, a^{3} b^{12} x^{6} + \frac{2730}{11} \, a^{4} b^{11} x^{\frac{11}{2}} + \frac{3003}{5} \, a^{5} b^{10} x^{5} + \frac{10010}{9} \, a^{6} b^{9} x^{\frac{9}{2}} + \frac{6435}{4} \, a^{7} b^{8} x^{4} + \frac{12870}{7} \, a^{8} b^{7} x^{\frac{7}{2}} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{6006}{5} \, a^{10} b^{5} x^{\frac{5}{2}} + \frac{1365}{2} \, a^{11} b^{4} x^{2} + \frac{910}{3} \, a^{12} b^{3} x^{\frac{3}{2}} + 105 \, a^{13} b^{2} x + a^{15} \log \left (x\right ) + 30 \, a^{14} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38106, size = 471, normalized size = 2.3 \begin{align*} \frac{15}{7} \, a b^{14} x^{7} + \frac{455}{6} \, a^{3} b^{12} x^{6} + \frac{3003}{5} \, a^{5} b^{10} x^{5} + \frac{6435}{4} \, a^{7} b^{8} x^{4} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{1365}{2} \, a^{11} b^{4} x^{2} + 105 \, a^{13} b^{2} x + 2 \, a^{15} \log \left (\sqrt{x}\right ) + \frac{2}{45045} \,{\left (3003 \, b^{15} x^{7} + 363825 \, a^{2} b^{13} x^{6} + 5589675 \, a^{4} b^{11} x^{5} + 25050025 \, a^{6} b^{9} x^{4} + 41409225 \, a^{8} b^{7} x^{3} + 27054027 \, a^{10} b^{5} x^{2} + 6831825 \, a^{12} b^{3} x + 675675 \, a^{14} b\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.51907, size = 211, normalized size = 1.03 \begin{align*} a^{15} \log{\left (x \right )} + 30 a^{14} b \sqrt{x} + 105 a^{13} b^{2} x + \frac{910 a^{12} b^{3} x^{\frac{3}{2}}}{3} + \frac{1365 a^{11} b^{4} x^{2}}{2} + \frac{6006 a^{10} b^{5} x^{\frac{5}{2}}}{5} + \frac{5005 a^{9} b^{6} x^{3}}{3} + \frac{12870 a^{8} b^{7} x^{\frac{7}{2}}}{7} + \frac{6435 a^{7} b^{8} x^{4}}{4} + \frac{10010 a^{6} b^{9} x^{\frac{9}{2}}}{9} + \frac{3003 a^{5} b^{10} x^{5}}{5} + \frac{2730 a^{4} b^{11} x^{\frac{11}{2}}}{11} + \frac{455 a^{3} b^{12} x^{6}}{6} + \frac{210 a^{2} b^{13} x^{\frac{13}{2}}}{13} + \frac{15 a b^{14} x^{7}}{7} + \frac{2 b^{15} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12505, size = 221, normalized size = 1.08 \begin{align*} \frac{2}{15} \, b^{15} x^{\frac{15}{2}} + \frac{15}{7} \, a b^{14} x^{7} + \frac{210}{13} \, a^{2} b^{13} x^{\frac{13}{2}} + \frac{455}{6} \, a^{3} b^{12} x^{6} + \frac{2730}{11} \, a^{4} b^{11} x^{\frac{11}{2}} + \frac{3003}{5} \, a^{5} b^{10} x^{5} + \frac{10010}{9} \, a^{6} b^{9} x^{\frac{9}{2}} + \frac{6435}{4} \, a^{7} b^{8} x^{4} + \frac{12870}{7} \, a^{8} b^{7} x^{\frac{7}{2}} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{6006}{5} \, a^{10} b^{5} x^{\frac{5}{2}} + \frac{1365}{2} \, a^{11} b^{4} x^{2} + \frac{910}{3} \, a^{12} b^{3} x^{\frac{3}{2}} + 105 \, a^{13} b^{2} x + a^{15} \log \left ({\left | x \right |}\right ) + 30 \, a^{14} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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